Consider the graph to the right. Explain the idea of a critical value. Then determine which x-values are critical values, and state why. A coordinate plane has a horizontal x-axis and a vertical y-axis. From left to right, ten points on the x-axis are l
Can you help me understand this Calculus question?
| Consider the graph to the right. Explain the idea of a critical value. Then determine whichx-values are critical values, and state why. |
A coordinate plane has a horizontal x-axis and a vertical y-axis. From left to right, ten points on the x-axis are labeled from x 1 to x 10. The graph of a function is a continuous curve in quadrant one, and its behavior from left to right is as follows. It falls to the bottom of a U shape at x 1. Then it rises, passing through a point at x 2, to the top of an inverted U shape at x 3. Then it falls to a cusp at x 4 that is higher than the point at x 1. Then it rises to the top of an inverted U shape at x 5 that is lower than the point at x 3. Then it falls to the bottom of a U shape at x 6 that is higher than the point at x 1 but lower than the point at x 4. Then it rises to a cusp at x 7 that is lower than the point at x 4. Then it falls to the bottom of a U shape at x 8 that is lower than the point at x 1. Then it rises, passing through a point at x 9, to the top of an inverted U shape at x 10 that is higher than the point at x 3. Then it falls. |
What is the idea of a critical value?
Critical values are x-values where relative extrema may occur.
Critical values are y-values where relative extrema may occur.
Critical values are the y-values of relative extrema.
Critical values are x-values where relative extrema occur.
Which x-values are critical values?
x 1x1,
x 3x3,
x 5x5,
x 6x6,
x 8x8,
and
x 10x10
because relative extrema occur at these points.
x 4x4
and
x 7x7
because a corner exists at these x-values.
x 1x1,
x 3x3,
x 4x4,
x 5x5,
x 6x6,
x 7x7,
x 8x8,
and
x 10x10
because the slopes of the tangent lines at these points are 0 or do not occur.
All x-values are critical values because they all occur at relative extrema or where the slopes of the tangent lines do not exist.
better explanation in screenshot 🙂
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